Kernel de calor

オベリスクの巨神兵

オベリスクの巨神兵

Miembro Maestro
[imath]\displaystyle\mathbb{P}_x\left\{\tau_C>t\right\} = \int^\infty_0\int_D{r^{n-1} \textcolor{blue}{\underbrace{\frac{e^{-\frac{\left(\rho^2+r^2\right)}{2t}}}{t(\rho r)^{\frac{n}{2}-1}}\sum^\infty_{j=1}I_{\alpha_j}\left( \frac{\rho r}{t} \right)m_j(\theta)m_j(\eta)}_{\text{Kernel de calor en el cono}}} \mathrm d\sigma(\eta)\mathrm dr}[/imath]
 
[imath] \displaystyle\mathbb{P}_x\left\{\tau_C>t\right\} = \int^\infty_0\int_D{r^{n-1} \left\{P^C_t(x,y)\right\} \mathrm d\sigma(\eta)\mathrm dr} = \int^\infty_0\int_D{\cancel{r^{n-1}} \frac{1}{\cancel{r^{n-1}}}P^n_t(\rho,r)\mathbb E_\rho \left[ \left.P^D_{T(t)}(\theta,\eta)\;\right\vert R_t=r\right]\mathrm d\sigma(\eta)\mathrm dr} = \int^\infty_0\int_D{P^n_t(\rho,r)\mathbb E_\rho \left[ \left.P^D_{T(t)}(\theta,\eta)\;\right\vert R_t=r\right]\mathrm d\sigma(\eta)\mathrm dr} [/imath]
 
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